A conjecture on B-groups

In this note, I propose the following conjecture~: a finite group G is nilpotent if and only if its largest quotient B-group \beta(G) is nilpotent. I give a proof of this conjecture under the additional assumption that G be solvable. I also show that this conjecture is equivalent to the following~: the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor.